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A Cyclic Group is a group that is generated by an element
. That is all the elements in are of the form , . We denote the cyclic group of order as . -
(Fraleigh e4.55 , Fraleigh 6.1) Every cyclic group is an Abelian Group.
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(Fraleigh 6.6) Every Subgroup of a cyclic group is Abelian
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(Fraleigh 6.7) The subgroups of
under addition are precisely the groups under addition for . This allows us to define the greatest common divisor, which is the positive generator of the cyclic group under addition.
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(Fraleigh 6.10)
is isomorphic to the integers under addition modulo . -
(Fraleigh 6.10) The infinite cyclic group is isomorphic to the integers under addition.
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(Fraleigh 6.14) If
, then where . -
(Fraleigh 6.16) The generators of a finite cyclic group
are of the form , where . ( and are coprime) -
(Fraleigh 10.11) Every group of prime order is cyclic.
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(Fraleigh 11.5) The group
is cyclic and is isomorphic to if and only if and are coprime. (Fraleigh 11.6) In general, the group
is cyclic and isomorphic to if and only if the between any pair of distinct ’s is .